Such waste waters are produced in large amounts, in particular, in mining during transport, refining and extraction of metal ores. Conventionally, sulfate is precipitated out of such waste waters as calcium sulfate by pH reduction and addition of a precipitant such as milk of lime, and removed from the waste water using sedimentation tanks or filter stages. Owing to the heavy metal content of the waste waters, the sulfate sludges must be treated as special waste and stored in special landfills at high expense. U.S. Pat. No. 5,587,079 discloses a method in which the heavy metals are precipitated out by introducing hydrogen sulfide and are separated off. Some of the sulfate-containing water is fed to a bioreactor in which sulfate is converted to H2S using sulfate-reducing bacteria, which H2S is used for the precipitation of heavy metals. Excess sulfate is converted to H2S in the bioreactor, wherein this is reacted to form metal sulfides. A disadvantage of this method is, especially, that the total amount of waste water must be passed through the bioreactor in order to separate off all of the sulfate contained therein. This requires not only a very large bioreactor having corresponding operating and service expenditure, but in addition further method steps and corresponding apparatuses for reacting the hydrogen sulfide generated to form metal sulfides. A further disadvantage is that in the case of large amounts of sulfate, a correspondingly large amount of metal sulfides is formed which are in addition to the metal sulfides which are precipitated out of the waste water containing heavy metals. Therefore, in total a very large amount of metal sulfides is produced which, in the lack of possibilities for reutilization lead to considerable problems of disposal and/or landfill. For instance, there is the risk, for example, that on contact with acidic water, hydrogen sulfide is expelled from the sulfides.